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 seminar class Active In SP Posts: 5,361 Joined: Feb 2011 12-03-2011, 12:10 PM   QUANTUM COMPUTING.doc (Size: 413.5 KB / Downloads: 181) CHAPTER 1 INTRODUCTION 1.1 GENERAL INTRODUCTION Quantum computing is a combination of physics, mathematics and computer science. Quantum algorithm exponentially “speed up” classical computation. The basic paradigm for quantum algorithm is the quantum circuit model, which is composed of the basic quantum units of information(qubits) and quantum gates. By interacting with each other while being isolated from the external environment, qubits can perform certain calculations exponentially faster than conventional computers. The quantum computer, following the laws of quantum physics, would gain enormous processing power through the ability to be in multiple states, and to perform tasks using all possible permutations simultaneously. By doing a computation on many different numbers at once, then interfering the results to get a single answer, a quantum computer has the potential to be much more powerful than a classical computer of the same size. 1.2 QUANTUM ALGORITHMS The main quantum algorithms are:  Quantum circuit based algorithm The Deutsch Oracle The Deutsch Jozsa Oracle The Simon Oracle Shor’s Algorithm Grover’s Algorithm  Adiabatic algorithm  Measurement based algorithm  Topological quantum field theory(TQFT) algorithm 1.3 ENEMIES OF QUANTUM COMPUTING There are two known enemies of quantum computing: a) Decoherence If we keep on putting quantum gates together into circuits we will quickly run into some serious practical problems. The more interacting qubits are involved the harder it tends to be to engineer the interaction that would display the quantum interference. Apart from the technical difficulties of working at single-atom and single-photon scales, one of the most important problems is that of preventing the surrounding environment from being affected by the interactions that generate quantum superposition. The more components the more likely it is that quantum computation will spread outside the computational unit and will irreversibly dissipate useful information to the environment. This process is called “Decoherence”. Even though we try to isolate the quantum system from the environment much as we can, we cannot supply total isolation. Therefore, the interaction of the quantum system and the environment result in “Decoherence” of the quantum state, which is equivalent to a partial measurement of the state by the environment. b) Gate Inaccuracies Decoherence is not the only problem with quantum computing. Gates, whether they are classical or quantum, are not perfect. The gates are usually combined together. So small errors in gates can combine together during computation and eventually causing failure, and it is not clear how to correct these small errors. The simplest example of error correcting code is a repetition code: replacing the bit we want to protect by 3 copies of the bit, 0 → (000) 1→ (111) Now an error may occur that causes one of the three bits to flip; If it’s the first bit, say, (000) → (100) (111) → (011) Now in spite of the error, the bit can be encoded correctly, by majority voting. CHAPTER 2 CONCEPTS OF QUANTUM COMPUTING 2.1 ELEMENTS OF QUANTUM COMPUTING Generally we’ll think of a quantum computer as a classical computer with a quantum circuit attached to it with some kind of interface between conventional and quantum logic. Since there are only a few things a quantum computer does better than a classical computer it makes sense to do the bulk of the processing on the classical machine. 1) Bits and Qubits These are the”nuts and bolts” of quantum computing. It describes qubits, gates, and circuits. Quantum computers perform operations on qubits which are analogous to conventional bits but they have an additional property in that they can be in a superposition. A quantum register with 3 qubits can store 8 numbers in superposition simultaneously, and a 250 qubit register holds more numbers (superposed) than there are atoms in the universe. Representation of data-qubits 2) Single Qubit Classical computers use two discrete states to represent a unit of information, this state is called a binary digit (or bit for short). A bit has the following two values: 0 and 1 There is no intermediate state between them, i.e. the value of the bit cannot be in a superposition. Quantum bits, or qubits, can on the other hand be in a state ”between” 0 and 1, but only during the computational phase of a quantum operation. When measured, a qubit can become either: The | > symbolic notation is part of the Dirac notation. 3) Multiple Qubit The potential amount of information available during the computational phase grows exponentially with the size of the system, i.e. the number of qubits. This is because if we have n qubits the number of basis states is 2n. E.g. if we have two qubits, forming a quantum register then there are four (=22) computational basis states: forming Here |01> means that qubit 1 is in state |0> and qubit 2 is in state |1>, etc. 2.2 CONCEPTS OF QUANTUM COMPUTING The following concepts are important for quantum computing: 1) Superposition Superposition means a system can be in two or more of its states simultaneously. For example a single particle can be traveling along two different paths at once. This implies that the particle has wave-like properties, which can mean that the waves from the different paths can interfere with each other. Interference can cause the particle to act in ways that are impossible to explain without these wave-like properties. The ability for the particle to be in a superposition is where we get the parallel nature of quantum computing: If each of the states corresponds to a different value then, if we have a superposition of such states and act on the system, we effectively act on all the states simultaneously. 2) Entanglement In 1935 Einstein (along with colleagues Podolski and Rosen) demonstrated a paradox (named EPR after them) in an attempt to refute the undefined nature of quantum systems. The results of their experiment seemed to show that quantum systems were defined, having local state BEFORE measurement. Although the original hypothesis was later proven wrong (i.e. it was proven that quantum systems do not have local state before measurement). The effect they demonstrated was still important, and later became known as entanglement. Entanglement is the ability for pairs of particles to interact over any distance instantaneously. Particles don’t exactly communicate, but there is a statistical correlation between results of measurements on each particle that is hard to understand using classical physics. To become entangled, two particles are allowed to interact; they then separate and, on measuring say, the velocity of one of them (regardless of the distance between them), we can be sure of the value of velocity of the other one (before it is measured). The reason we say that they communicate instantaneously is because they store no local state and only have well defined state once they are measured. Because of this limitation particles can’t be used to transmit classical messages faster than the speed of light as we only know the states upon measurement. Entanglement has applications in a wide variety of quantum algorithms and machinery. 3) Uncertainty The quantum world is irreducibly small so it’s impossible to measure a quantum system without having an effect on that system as our measurement device is also quantum mechanical. As a result there is no way of accurately predicting all of the properties of a particle. There is a trade off - the properties occur in complementary pairs (like position and momentum, or vertical spin and horizontal spin) and if we know one property with a high degree of certainty then we must know almost nothing about the other property. That unknown property’s behaviour is essentially random. An example of this is a particle’s position and velocity: if we know exactly where it is then we know nothing about how fast it is going. This indeterminacy is exploited in quantum cryptography.
 seminar class Active In SP Posts: 5,361 Joined: Feb 2011 16-03-2011, 02:09 PM presented by: Joseph Stelmach   quantumComputers.ppt (Size: 223.5 KB / Downloads: 214) Quantum Computing What is a quantum computer?  A quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level.  “I think I can safely say that nobody understands quantum mechanics” - Feynman  1982 - Feynman proposed the idea of creating machines based on the laws of quantum mechanics instead of the laws of classical physics.  1985 - David Deutsch developed the quantum turing machine, showing that quantum circuits are universal.  1994 - Peter Shor came up with a quantum algorithm to factor very large numbers in polynomial time.  1997 - Lov Grover develops a quantum search algorithm with O(√N) complexity Representation of Data - Qubits A bit of data is represented by a single atom that is in one of two states denoted by |0> and |1>. A single bit of this form is known as a qubit A physical implementation of a qubit could use the two energy levels of an atom. An excited state representing |1> and a ground state representing |0>. Data Retrieval  In general, an n qubit register can represent the numbers 0 through 2^n-1 simultaneously. Sound too good to be true?…It is!  If we attempt to retrieve the values represented within a superposition, the superposition randomly collapses to represent just one of the original values. Relationships among data – Entanglement  Entanglement is the ability of quantum systems to exhibit correlations between states within a superposition.  Imagine two qubits, each in the state |0> + |1> (a superposition of the 0 and 1.) We can entangle the two qubits such that the measurement of one qubit is always correlated to the measurement of the other qubit. Quantum Gates  Quantum Gates are similar to classical gates, but do not have a degenerate output. i.e. their original input state can be derived from their output state, uniquely. They must be reversible.  This means that a deterministic computation can be performed on a quantum computer only if it is reversible. Luckily, it has been shown that any deterministic computation can be made reversible.(Charles Bennet, 1973) Quantum Gates – Hadamard Simplest gate involves one qubit and is called a Hadamard Gate (also known as a square-root of NOT gate.) Used to put qubits into superposition
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